What is a Square Root?
The square root of a number n is a value that, when multiplied by itself, gives the original number n. It is written using the radical symbol √ or as n1/2.
Example: √144 = 12 because 12 × 12 = 144
Every positive number has two square roots — one positive and one negative. For example, both +12 and −12 are square roots of 144. However, the principal square root (the one our calculator returns) is always the non-negative value.
The square root of 0 is 0. Square roots of negative numbers are not real numbers — they belong to the complex number system (written using the imaginary unit i).
Perfect Squares — Memorise These
A perfect square is a number whose square root is a whole integer. Memorising these saves enormous time in exams and everyday calculations.
| Number (n) | Square Root (√n) | Number (n) | Square Root (√n) |
|---|---|---|---|
| 1 | 1 | 100 | 10 |
| 4 | 2 | 121 | 11 |
| 9 | 3 | 144 | 12 |
| 16 | 4 | 169 | 13 |
| 25 | 5 | 196 | 14 |
| 36 | 6 | 225 | 15 |
| 49 | 7 | 256 | 16 |
| 64 | 8 | 289 | 17 |
| 81 | 9 | 324 | 18 |
How to Calculate Square Root (Long Division Method)
Before calculators existed, mathematicians used the long division method to find square roots by hand. While our calculator does this instantly, understanding the manual method deepens your mathematical insight.
Step-by-step example: √529
- Group digits from right in pairs: 5 | 29
- Find largest integer whose square ≤ 5: that's 2 (2²=4). Quotient starts with 2.
- Subtract: 5−4 = 1. Bring down next pair: 129.
- Double the quotient (2×2=4). Find digit d so that (40+d)×d ≤ 129: d=3 → 43×3=129 ✓
- √529 = 23
Square Root Properties and Rules
Understanding the algebraic properties of square roots helps simplify complex expressions in algebra and calculus.
√(a / b) = √a / √b (Quotient rule)
√(a²) = |a| (Absolute value)
(√a)² = a (Square cancels root)
√a + √b ≠ √(a+b) (Cannot be simplified this way)
Simplifying square roots: To simplify √72, find the largest perfect square factor: 72 = 36 × 2, so √72 = √36 × √2 = 6√2 ≈ 8.485.
Real-World Applications of Square Roots
Square roots are not just academic — they appear in physics, engineering, architecture, finance, and everyday geometry.
- Pythagorean Theorem: To find the hypotenuse c of a right triangle: c = √(a² + b²). A 3-4-5 triangle: c = √(9+16) = √25 = 5.
- Distance Formula: The distance between two points (x₁,y₁) and (x₂,y₂): d = √((x₂−x₁)² + (y₂−y₁)²).
- Physics — Free Fall: Time to fall height h: t = √(2h/g). For h=20m: t = √(40/9.8) ≈ 2.02 seconds.
- Finance — Volatility: Standard deviation (risk) uses square roots of variance in portfolio analysis.
- Architecture: Finding diagonal dimensions of rooms, screens, fields, and buildings.
- Signal Processing: RMS (Root Mean Square) values of AC electrical signals use square roots.